## Description

Definition and examples of vector spaces (over arbitrary fields), subspaces, direct sum of subspaces, linear dependence and independence, basis and dimensions, linear transformations, quotient spaces, algebra of linear transformations, linear functions, dual spaces, matrix representation of a linear transformation, rank and nullity of a linear transformation, invariant subspaces.

Characteristic polynomial and minimal polynomial of a linear transformation, eigen values and eigen vectors of a linear transformation, diagonalization and triangularization of a matrix, Jordan and Rational canonical forms, bilinear forms, symmetric bilinear forms, Sylvester’s theorem, quadratic forms, Hermitian forms, Inner product spaces, Gram-schmidt orthonormalization process.